I am trying to find out when equality holds in Minkowski's inequality for $L^{\infty}$ (i.e. a necessary and sufficient condition for equality). I did a search and there was a discussion for the case where $1<p<\infty$ but not when $p=\infty$ so I am hoping to get some ideas or for someone to point me to a source where this is discussed.

I will list a couple of observations I made while working this out (though I'm not sure whether I'm right with these):

If $\mu(\{x:|f(x)|\geq||f+g||_{\infty}-||g||_{\infty}\})=0$ (or with $f$ and $g$ interchanged), then I have the reverse inequality.

If I pick $a,b$ such that $||f||_{\infty}\leq a<||f||_{\infty}+\varepsilon$, $||g||_{\infty}\leq b<||g||_{\infty}+\varepsilon$, $\mu(\{x:|f(x)|>a\})=0$, $\mu(\{x:|g(x)|>b\})=0$, and for all $c<a+b$ I have $\mu(\{x:|f(x)+g(x)|>c\})>0$, then I also have the reverse inequality.